Thermal Rayleigh-Marangoni convection in a three-layer liquid-metal-battery model

The combined effects of buoyancy-driven Rayleigh-Bénard convection (RC) and surface tension-driven Marangoni convection (MC) are studied in a triple-layer configuration which serves as a simplified model for a liquid metal battery (LMB). The three-layer model consists of a liquid metal alloy cathode, a molten salt separation layer, and a liquid metal anode at the top. Convection is triggered by the temperature gradient between the hot electrolyte and the colder electrodes, which is a consequence of the release of resistive heat during operation. We present a linear stability analysis of the state of pure thermal conduction in combination with three-dimensional direct numerical simulations of the nonlinear turbulent evolution on the basis of a pseudospectral method. Five different modes of convection are identified in the configuration, which are partly coupled to each other: RC in the upper electrode, RC with internal heating in the molten salt layer, and MC at both interfaces between molten salt and electrode as well as anticonvection in the middle layer and lower electrode. The linear stability analysis confirms that the additional Marangoni effect in the present setup increases the growth rates of the linearly unstable modes, i.e., Marangoni and Rayleigh-Bénard instability act together in the molten salt layer. The critical Grashof and Marangoni numbers decrease with increasing middle layer thickness. The calculated thresholds for the onset of convection are found for realistic current densities of laboratory-sized LMBs. The global turbulent heat transfer follows scaling predictions for internally heated RC. The global turbulent momentum transfer is comparable with turbulent convection in the classical Rayleigh-Bénard case. In summary, our studies show that incorporating Marangoni effects generates smaller flow structures, alters the velocity magnitudes, and enhances the turbulent heat transfer across the triple-layer configuration.

Figure 1

Sketch of the simplified three-layer liquid-metal battery model. All size lengths are given in units of the lower layer height $d^{\left(1\right)}$.

Figure 2

The stationary temperature profile of the reference case with the parameters taken from Table 2. In the pure conduction state one sets $\text{Ma}=G=0$. The profile is a plot of the solution (35, 36, 37). The profile is compared with the solution of a direct numerical simulation of the Boussinesq equations at the same parameters.

Figure 3

Linear stability analysis of the pure Marangoni convection (MC) case at $G=0$ for the reference parameters of Table 2. (a) Neutral stability curve $\mathrm{Ma}\left(\mathrm{k}\right)$. (b) Root mean square velocity in the middle layer for simulation runs LS01 and LS02.

Figure 4

Comparison of eigenmodes from stability analysis and full numerical simulation. (a) The difference of the profiles of temperature, $T(x=0,y=0,z,t=10)-T(x=0,y=0,z,t=3)$, is plotted. Values from the eigenvalue calculation with $k=2.15$ and the simulation LS02 are shown. (b) The difference of the profiles of vertical velocity component, $u_z(x=0,y=0,z,t=10)-u_z(x=0,y=0,z,t=3)$ is displayed. Again, values from the eigenvalue calculation with $k=2.15$ and the simulation LS02 are shown.

Figure 5

Linear stability analysis of Rayleigh-Marangoni convection (RMC) case at $\mathrm{Ma}\ne 0$ and $G\ne 0$. (a) Critical Marangoni number ${\mathrm{Ma}}_c$ as a function of the Grashof number $G$. (b) Critical wave number corresponding to the analysis in the top panel.

Figure 6

Eigenfunction profiles across the full battery height for pure Rayleigh-Bénard convection at $\mathrm{Ma}=0$, ${\tilde{G}}_c=-1.2937\times {10}^4$, and $k_c=3$, respectively.

Figure 7

Linear stability analysis of Marangoni convection (MC) for varying middle layer heights $d_{21}$. The compensated critical Marangoni number ${\widetilde{\mathrm{Ma}}}_c{d}_{21}$ as a function of the middle layer height $d_{21}$ is displayed.

Figure 8

Plot of the compensated critical Grashof number ${\tilde{G}}_c{\left(d_{21}\right)}^3$ as a function of the middle layer height $d_{21}$.

Figure 9

Eigenfunctions at $\mathrm{Ma}=0$ of velocity perturbation $w\left(z\right)$ for three midlayer heights corresponding to the critical parameters. The relative thickness of the middle layer height is indicated in the legend.

Figure 10

Simulation of convection slightly above the threshold with $G=-1.33\times {10}^4$, which corresponds to the material parameters derived in Table 1 but a shallower layer of $d^{\left(i\right)}=6.4$ mm (simulation runs LS11 and LS12). Root-mean-square velocity in the middle layer as function of time is displayed.

Figure 11

Contour-vector plots of temperature and velocity of the simulations of convection near the threshold. (a) Simulation snapshot of LS12 with $\mathrm{Ma}=0$ at $t=30$ shows a vertical cut at $y=0$ with temperature contours and velocity field vectors. (b) Velocity and temperature for the same data at the upper interface II. (c, d) The same quantities as (a, b) for LS11 at $t=20$.

Figure 12

Convection in the reference configuration H4 of Table 4. Root mean square velocities in each of the three layers versus time are shown for RC (a), MC (b), and RMC (c).

Figure 13

Convection in the reference configuration H4 of Table 4. Vertical profiles of the $x,y$-averaged rms velocity in cases RC, MC, and RMC, respectively, are shown in panel (a). $x,y$-averaged temperatures are shown in panel (b). All data are also averaged in time. Furthermore, we plot the vertical temperature profile of the pure conduction state for comparison.

Figure 14

Convection in the reference configuration H4 of Table 4. Nusselt number Nu as defined in Eq. (68) versus time for runs RC, MC, and RMC. Line styles agree with those in Fig. 13.

Figure 15

Rayleigh-Bénard convection ($\mathrm{Ma}=0$) in the reference configuration H4. Temperature and velocity distributions at $t=0.3$ are shown. Temperature is displayed as a contour plot, velocity by arrows projected into the corresponding plane. (a) Temperature and velocity at the upper interface II. The length of the reference velocity arrows is always in units of $U_{\mathrm{vis}}$. (b) Temperature and velocity at the lower interface I. (c) Vertical cut at $y=0$. (d) Isosurface of temperature at $T=0.37$.

Figure 16

Marangoni convection $(G=0)$ in the reference configuration H4. See Fig. 15 for details. Here the temperature isosurface in (d) is taken at $T=0.5$.

Figure 17

Rayleigh-Marangoni convection in the reference configuration H4. See Fig. 15 for details. Here, the temperature isosurface in the lower right panel is taken at $T=0.37$ also.

Figure 18

Root-mean-square velocities as a function of the magnitude of the Grashof number, which is directly connected to the variation of the reference height $d^{\left(1\right)}$. Solid lines are for RMC and dashed lines for RC. Each of the three layers is indicated in the plot.

Figure 19

Global transport of heat (bottom panel) versus Grashof number magnitude for DNS series H1 to H6 as measured by the Nusselt number for the whole system (68).

Figure 20

Vertical instantaneous cut through the three-layer battery model for case H6 at $y=0$, $t=0.04$. Temperature field contours are plotted.

Figure 21

Vertical temperature gradient at the upper interface I. Contours of ${\partial }_z{T}^{\left(2\right)}(x,y,z=d_{21})$ for simulation runs with variable middle layer height are shown: (a) case H2 at $t=0.5$; (b) case H3 at $t=0.3$; (c) case H5 at $t=0.2$; (d) case H6 at $t=0.04$.

Figure 22

Reynolds number scaling versus Grashof number for shallower middle layers and different electrical current densities with RMC (solid line) and RC (dashed line). (a) Reynolds numbers ${\mathrm{Re}}^{\left(\mathrm{i}\right)}$ for series JM; (b) Reynolds number for series JS. As a guide to the eye, we also plot $\mathrm{Re}\sim {\left|G\right|}^{0.5}$ as a dash-dotted line.

Figure 23

Nusselt number scaling versus Grashof number for shallower middle layers and different electrical current densities. (a) Nusselt numbers for JM and JS series; (b) all Nusselt numbers replotted as a function of the relative Grashof number $G_r=G/{\tilde{G}}_c$.

Figure 24

Anticonvection observed in a simulation for parameters of JM4, but with $\mathrm{Ma}=0$ and ${\beta }_{21}={\beta }_{31}=0$. A vertical cut $y=0$ with velocity vectors and temperature contours at a time of $t=0.145$ is shown. This is shortly after the onset of anticonvection.